Lefschetz formulae and zeta functions

Searching symbolically for Apery-like formulae for values of the Riemann zeta function

We discuss some aspects of the search for identities using computer algebra and symbolic methods. The focus is on so-called Apery-like formulae for special values of the Riemann Zeta function. Much work lays ahead in formally proving and properly classifying many of the identities we have uncovered.

The summation of infinite partial fraction decomposition I: some formulae related to the Hurwitz zeta function

In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.

Geometric zeta functions for higher rank p-adic groups

The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions

Zeta functions of nondegenerate hypersurfaces in toric varieties via controlled reduction in $p$-adic cohomology

We give an interim report on some improvements and generalizations of the Abbott-Kedlaya-Roe method to compute the zeta function of a nondegenerate ample hypersurface in a projectively normal toric variety over $\mathbb{F}_p$ in linear time in $p$. These are illustrated with a number of examples including K3 surfaces, Calabi-Yau threefolds, and a cubic fourfold. The latter example is a non-special cubic fourfold appearing in the Ranestad-Voisin coplanar divisor on moduli space; this verifies that the coplanar divisor is not a Noether-Lefschetz divisor in the sense of Hassett.

Empirically determined Apery-like formulae for zeta(4n+3)

Some rapidly convergent formulae for special values of the Riemann zeta function are given. We obtain a generating function formula for zeta(4n+3) which generalizes Apery's series for zeta(3), and appears to give the best possible series relations of this type, at least for n<12. The formula reduces to a finite but apparently non-trivial combinatorial identity. The identity is equivalent to an interesting new integral evaluation for the central binomial coefficient. We outline a new technique for transforming and summing certain infinite series. We also derive a beautiful formula which provide…

Generalized Jacobi-Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi-Trudi formulae and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulae we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.

Taylor coefficients of Anderson-Thakur series and explicit formulae

For each positive characteristic multiple zeta value (defined by Thakur), the first and third authors constructed a $t$-module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value. The objective of this paper is to use the Taylor coefficients of Anderson-Thakur series and $t$-motivic Carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector.

A note on the fourth power moment of the Riemann zeta-function

We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.

多项式对数函数的Ramaswami类型翻译公式

Ramaswami Type translation formulae for the polylogarithm functions

本文研究多项式对数函数的Ramaswami和Apostol类型翻译公式，并探讨其在伯努利数递推关系和切线数同余性质中的应用。

On the Asymptotics to all Orders of the Riemann Zeta Function and of a Two-Parameter Generalization of the Riemann Zeta Function

We present several formulae for the large $t$ asymptotics of the Riemann zeta function $ζ(s)$, $s=σ+i t$, $0\leq σ\leq 1$, $t>0$, which are valid to all orders. A particular case of these results coincides with the classical results of Siegel. Using these formulae, we derive explicit representations for the sum $\sum_a^b n^{-s}$ for certain ranges of $a$ and $b$. In addition, we present precise estimates relating this sum with the sum $\sum_c^d n^{s-1}$ for certain ranges of $a, b, c, d$. We also study a two-parameter generalization of the Riemann zeta function which we denote by $Φ(u,v,β)$, $…

Asymptotics to all orders of the Hurwitz zeta function

We present several formulae for the large-$t$ asymptotics of the modified Hurwitz zeta function $ζ_1(x,s),x>0,s=σ+it,0<σ\leq1,t>0,$ which are valid to all orders. In the case of $x=0$, these formulae reduce to the asymptotic expressions recently obtained for the Riemann zeta function, which include the classical results of Siegel as a particular case.

Mixed moments of the Riemann zeta function

We analyse a collection of mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$ conjecture.

On shuffle-type functional relations of desingularized multiple zeta-functions

We treat desingularized multiple zeta-functions introduced by Furusho, Komori, Matsumoto and Tsumura. In this paper, we prove functional relations, which are shuffle type product formulae, between desingularized multiple zeta-functions and desingularized values.

Functional equations for local normal zeta functions of nilpotent groups

We give explicit formulae for the local normal zeta functions of torsion-free, class-2-nilpotent groups, subject to conditions on the associated Pfaffian hypersurface which are generically satisfied by groups with small centre and sufficiently large abelianization. We show how the functional equations of two types of zeta functions - the Weil zeta function associated to an algebraic variety and zeta functions of algebraic groups introduced by Igusa - match up to give a functional equation for local normal zeta functions of groups. We also give explicit formulae and derive functional equations …

Moments of the Riemann zeta-function

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments in families of $L$-functions.

Scaling group flow and Lefschetz trace formula for laminated spaces with $p-$adic transversal

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function $\hatζ(s)$ (resp. the zeta function $ζ_Y(s)$ of a smooth projective curve $Y$ over a finite field $\mathbb{F}_q$, $q=p^f$)) one could possibly associate a foliated Riemannian laminated space $(S_{\mathbb{Q}}, \mathcal{F}, g, ϕ^t)$ (resp. $(S_Y, \mathcal{F}, g, ϕ^t)$) endowed with an action of a flow $ϕ^t$ whose primitive compact orbits should correspond to the primes of $\mathbb{Q}$ (resp. $Y$). The existence of such a foliated space and flow $ϕ^t$ is still unknown except when $Y$ is an ellipt…

On values of the higher derivatives of the Barnes zeta function at non-positive integers

Let $x$ be a complex number which has a positive real part, and $w_1,\ldots,w_N$ be positive rational numbers. We show that $w^s ζ_N (s, x \ |\ w_1,\ldots, w_N)$ can be expressed as a finite linear combination of the Hurwitz zeta functions over $\mathbb Q(x)$, where $ζ_N (s,x \ |\ w_1,\ldots, w_N)$ is the Barnes zeta function and $w$ is a positive rational number explicitly determined by $w_1,\ldots, w_N$. Furthermore, we give generalizations of Kummer's formula on the gamma function and Koyama-Kurokawa's formulae on the multiple gamma functions, and an explicit formula for the values at non-p…

Twisted mixed moments of the Riemann zeta function

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a polynomial $P(x)$. Such examinations are performed both unconditionally and under the assumption of a weaker version of the $abc$-conjecture.

On the Gram's Law in the Theory of Riemann Zeta Function

Some statements concerning the distribution of imaginary parts of zeros of the Riemann zeta\,-function are established. These assertions are connected with so\,-called `Gram's law' or `Gram's rule'. In particular, we give a proof of several Selberg's formulae stated him without proof in his paper `The Zeta Function and the Riemann Hypothesis' (1946), and some of their equivalents.

关于临界线上 Riemann zeta 函数一阶矩的渐近公式

On Asymptotic Formula for the First Moment of the Riemann Zeta-Function on the Critical Line

本文在 Riemann 假设成立的前提下，证明了临界线上 Riemann zeta 函数一阶矩的渐近公式。

黎曼ζ函数零点序数的林德勒夫假设

The Lindelöf Hypothesis for Zeta Zero Ordinates

研究了非平凡零点序数的指数和渐进行为，探讨了林德勒夫假设的相关结论。

A Gamma Distribution Hypothesis for Prime $k$-tuples

We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates. Possessing average counting functions along with their corresponding exact counting functions allows to implicitly define pertinent $k$-tuple zeta functions. The $k$-tuple zeta functions in turn allow construction of $k$-tuple analogs of explicit formulae. If the zeros of the (implicitly defined) $k$-tuple zeta can be determined, the explicit formulae should yield…

Hurwitz Zeta 函数在正整数点的值

The Hurwitz Zeta Function at the Positive Integers

本文通过分别求解实部和虚部并合并，得到 Hurwitz Zeta 函数在正整数 k 处的公式，并给出了生成函数的解析延拓。

The alpha-calculus-cum-alpha-analysis of r-th order derivative of zeta(s,alpha)

For the higher order derivative(with respect to the first variable) of Hurwitz zeta function,we discuss as a function of the second variable,the location and the nature of its singularities and obtain the formulae for its derivative and primitive and also discuss its Riemann integrability on the unit interval and on the interval[1,infinity).We also evaluate certain integrals on the unit interval involving these functions.

曲线上点计数的复杂性以及曲面 zeta 函数的因子 $P_1(T)$

Complexity of counting points on curves and the factor $P_1(T)$ of the zeta function of surfaces

本文研究有限域上曲线和曲面点计数的计算复杂性，提出了首个高效的 Arthur-Merlin 协议证明曲线点数及相关结构，并扩展至曲面 zeta 函数因子 $P_1(T)$ 的计算，给出了首个非平凡的计算复杂性上界。

Upper bounds for the moments of zeta prime rho

Assuming the Riemann Hypothesis, we obtain an upper bound for the 2k-th moment of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$ for every positive integer k. Our bounds are nearly as sharp as the conjectured asymptotic formulae for these moments. The proof is based upon a recent method of K. Soundararajan that provides analogous bounds for continuous moments of the Riemann zeta-function as well as for moments L-functions at the central point, averaged over families.

Averages of ratios of the Riemann zeta-function and correlations of divisor sums

We establish a connection between the ratios conjecture for the Riemann zeta-function and a conjecture concerning correlations of convolutions of Möbius and divisor functions. Specifically, we prove that the ratios conjecture and an arithmetic correlations conjecture imply the same result. This provides new support for the ratios conjecture, which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe. Our main theorem generalises a recent calculation pertaining to the special case of two-over-two ratios.

Zeta functions of groups and enumeration in Bruhat-Tits buildings

We introduce a new method to calculate local normal zeta functions of finitely generated, torsion-free nilpotent groups. It is based on an enumeration of vertices in the Bruhat-Tits building for Sl_n(Q_p). It enables us to give explicit computations for groups of class 2 with small centres and to derive local functional equations. Examples include formulae for non-uniform normal zeta functions.

关于广义双重级数的一个注记

A note on a generalized double series

本文利用围道积分推导出涉及 Hurwitz-Lerch zeta 函数的广义双重有限级数的闭式公式，并求出其用特殊函数表示的特例。

二维半局部环上的 Bushnell-Reiner zeta 函数

Bushnell-Reiner zeta functions over two-dimensional semilocal rings

本文基于可逆理想过滤方法，将交换二维正则局部环的 zeta 函数无穷乘积公式推广至非交换情形，并给出了 Hey 公式的二维类比及 Rump 环上的有效公式。

Zeta functions over zeros of general zeta and $L$-functions

We describe in detail three distinct families of generalized zeta functions built over the (nontrivial) zeros of a rather general arithmetic zeta or L-function, extending the scope of two earlier works that treated the Riemann zeros only. Explicit properties are also displayed more clearly than before. Several tables of formulae cover the simplest concrete cases: L-functions for real primitive Dirichlet characters, and Dedekind zeta functions.

黎曼zeta函数在等差数列上的均方值

Mean-square values of the Riemann zeta function on arithmetic progressions

本文研究了黎曼zeta函数在等差数列上的二阶离散矩，推导出其渐近公式，揭示了离散矩与连续矩的关系，并指出当a为正整数时主项相等。

黎曼ζ函数在直线$\Re(s) = 1$和奇数参数上的评估

Evaluation of Riemann Zeta function on the Line $\Re(s) = 1$ and Odd Arguments

本文针对黎曼ζ函数在奇数参数和直线$\Re(s) = 1$上的评估，提出了一个具有指数收敛性的简单近似公式，并与文献中的其他公式进行了比较。

The Riemann zeta in terms of the dilogarithm

We give a representation of the classical Riemann $ζ$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen $Gl_2$-function). We also derive corresponding representations involving the derivatives of the $Gl_2$-function. A generalized symmetrized Müntz-type formula is also derived. For a special choice of test functions it connects to our integral representation of the $ζ$-function, providing also a computation of a concrete Mellin transform. Certain formulae involving seri…

Enumerating submodules invariant under an endomorphism

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence $2$.

Special zeta Mahler functions

In 1969, I. Bernstein and S. Gelfand introduced an object, which is now called the zeta Mahler function (ZMF, also zeta Mahler measure) and related to the Mahler measure. Here we discuss a family of ZMFs attached to the Laurent polynomials $k + (x_1 + x_1^{-1}) \cdots (x_r + x_r^{-1})$, where $k$ is real. We give explicit formulae, present examples and establish properties for these ZMFs, such as an RH-type phenomenon. Further, we explore relations with the Mahler measure.

Unconditional and Conditional Large Gaps between the zeros of the Riemann Zeta-Function

In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) time…

Riemann zeta函数的放大矩

Amplified moments of the Riemann zeta function

建立了Riemann zeta函数的两片放大第二和第四矩的渐近公式，并应用于无条件获得zeta的若干联合矩的有效下界，与Keating-Wei和Keating-Snaith的猜想高度一致。

多重zeta值与Drinfeld结合子级数的delta值之间的关系

Relations between multiple zeta values and delta values from Drinfeld's associator series

该研究通过比较Drinfeld结合子的两个级数公式，推导得出多重zeta值与delta值的新关系，并从五阶项中发现两个新关系。

Flag Hilbert-Poincaré series of hyperplane arrangements and their Igusa zeta functions

We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincaré series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ i…

Moments of the derivative of the Riemann zeta-function and of characteristic polynomials

We investigate the moments of the derivative, on the unit circle, of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the Riemann zeta-function on the critical line. We do the same for the analogue of Hardy's Z-function, the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle. Our formulae are expressed in terms of a determinant of a matrix whose entries involve the I-Bessel function and, alternately, by a combinatorial sum.

On the Enumeration of Irreducible Polynomials over $\text{GF}(q)$ with Prescribed Coefficients

We present an efficient deterministic algorithm which outputs exact expressions in terms of $n$ for the number of monic degree $n$ irreducible polynomials over $\mathbb{F}_{q}$ of characteristic $p$ for which the first $l < p$ coefficients are prescribed, provided that $n$ is coprime to $p$. Each of these counts is $\frac{1}{n}(q^{n-l} + \mathcal{O}(q^{n/2}))$. The main idea behind the algorithm is to associate to an equivalent problem a set of Artin-Schreier curves defined over $\mathbb{F}_q$ whose number of $\mathbb{F}_{q^n}$-rational affine points must be combined. This is accomplished by c…

Jacob 阶梯以及与 Hardy-Littlewood-Ingham（1926）公式和 Ingham（1926）公式的某种杂交相关的 Fermat-Wiles 定理的新等价命题

Jacob's ladders and new equivalents of the Fermat-Wiles theorem connected with some cross-bred of the formulae of Hardy-Littlewood-Ingham (1926) and of Ingham (1926)

该研究通过建立临界线与临界带中 ζ-积分之间的新公式，并杂交两类 Ingham 相关公式，给出了 Fermat-Wiles 定理的新 ζ-等价命题。

On A Rapidly Converging Series For The Riemann Zeta Function

To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates…

Large Spaces Between the Zeros of the Riemann Zeta-Function

In this paper, we will employ the Opial and Wirtinger type inequalities to derive some conditional and unconditional lower bounds for the gaps between the zeros of the Riemann zeta-function. First, we prove (unconditionally) that the consecutive nontrivial zeros often differ by at least 1.9902 times the average spacing. This value improves the value 1.9 due to Mueller and the value 1.9799 due to Montogomery and Odlyzko. Second, on the hypothesis that the 2k-th mixed moments of the Hardy Z-function and its derivative are correctly predicted by random matrix theory, we derive some explicit formu…

扩展的Levett三角级数

Extended Levett trigonometric series

本文通过研究含3的幂次等比数列角度的两类扩展有限三角级数，推导了涉及Hurwitz-Lerch zeta函数的闭式公式，并以此得到含特殊函数、三角函数与基本常数的复合有限级数，还给出了结果简表。

Barnes多重zeta函数的Laurent级数展开

Laurent series expansions for the Barnes multiple zeta function

本文研究Barnes多重zeta函数的Laurent系数，在最高极点给出显式极限公式，在较低极点建立与Taylor系数的全阶关系，并利用整函数性质分析大阶行为，推广了常数项的有限部分表示。

Bandlimited approximations and estimates for the Riemann zeta-function

In this paper, we provide explicit upper and lower bounds for the argument of the Riemann zeta-function and its antiderivatives in the critical strip under the assumption of the Riemann hypothesis. This extends the previously known bounds for these quantities on the critical line (and sharpens the error terms in such estimates). Our tools come not only from number theory, but also from Fourier analysis and approximation theory. An important element in our strategy is the ability to solve a Fourier optimization problem with constraints, namely, the problem of majorizing certain real-valued even…

The 4.36-th moment of the Riemann zeta-function

Conditionally on the Riemann Hypothesis we obtain bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta-function for all positive real k < 2.181. This provides for the first time an upper bound of the correct order of magnitude for some k > 2; the case of k = 2 corresponds to a classical result of Ingham. We prove our result by establishing a connection between moments with k > 2 and the so-called "twisted fourth moment". This allows us to appeal to a recent result of Hughes and Young. Furthermore we obtain a point-wise bound for |zeta(1/2 + it)|^{2r} (with 0 < r < …

适用于整个复平面的ζ函数及其推广形式的最简积分

Simplest Integrals for the Zeta Function and its Generalizations Valid in All $\mathbb{C}$

本文扩展Faulhaber公式得到黎曼ζ函数公式，进而推导其推广形式的公式，所用方法涉及特定Lerch transcendent组合的极限计算，相关结果可用于求H_{-k}(n)的泰勒级数。

The arithmetic Kuznetsov formula on $GL(3)$, I: The Whittaker case

The original formulae of Kuznetsov for $SL(2,\mathbb{Z})$ allowed one to study either a spectral average via Kloosterman sums or to study an average of Kloosterman sums via a spectral interpretation. In previous papers, we have developed the spectral Kuznetsov formulae at the minimal weights for $SL(3,\mathbb{Z}))$, and in these formulae, the big-cell Kloosterman sums occur with weight functions attached to four different integral kernels, according to the choice of signs of the indices. These correspond to the $J$- and $K$-Bessel functions in the case of $GL(2)$. In this paper, we demonstrate…

关于CUE特征多项式导数与黎曼zeta函数的矩

On moments of the derivative of CUE characteristic polynomials and the Riemann zeta function

本文研究了N×N Haar分布酉矩阵特征多项式导数的复值矩，得到了显式公式，并探索其与远离临界线的黎曼zeta函数导数矩的联系，在Lindelöf假设下证明正整数矩与随机矩阵结果一致，同时提出非整数矩的渐近猜想。

CUE 特征多项式高阶导数联合矩 I：渐近公式

Joint moments of higher order derivatives of CUE characteristic polynomials I: asymptotic formulae

本文推导了CUE随机矩阵特征多项式任意阶导数联合矩的显式渐近公式，以行列式和组合和表示，并借此为Riemann zeta函数及Hardy的Z函数导数联合矩提出了受数据支持的普遍猜想。

关于复数上贝塞尔函数的汉克尔变换及高斯域上的显式谱公式

On the Hankel Transform of Bessel Functions on Complex Numbers and Explicit Spectral Formulae over the Gaussian Field

本文在复数域上证明贝塞尔函数的汉克尔-梅林与双重傅里叶-梅林变换积分公式，并应用于高斯数域上戴德金zeta函数四次矩及皮卡群中心L值二阶矩的显式谱公式，同时建立分布汉克尔变换理论。